Algebra Know-It-ALL

610 Worked-Out Solutions to Exercises: Chapters 1 to 9

repeating sequence in the numerator and a string of 16 digits, all 9s, in the denominator,

inserting commas to make the large numbers more readable. That gives us

0,588,235,294,117,647 / 9,999,999,999,999,999

The initial cipher can now be removed. It was only necessary to be sure we put the correct

number of 9s in the denominator. The final answer is:

588,235,294,117,647 / 9,999,999,999,999,999

If you want, you can check this by dividing it out using a calculator with a large display,

such as the one in a computer. You should get the same result as we got when we divided

1 by 17. Another way to check this is to divide the denominator of the above fraction by

the numerator (that is, take the quotient “upside-down”). You should get exactly 17.

8. The number we are given to start with is 2.892892892.... To convert this to a ratio of integers,
   we first write down the part of the expression to the right of the decimal point, like this:
   . 892 892 892 ...

From this, we know that fractional part of the expression is 892/999. We put back the

whole-number part, getting

2-892/999

Now we must convert 2 to a fraction with a denominator 999. We multiply 2 by 999,

getting 1,998. This goes into the numerator. Now we have two fractions that we can

easily add to produce the final ratio:

892 / 999 + 1,998 / 999 = (892 + 1,998) / 999

= 2,890 / 999

It is always a good idea to check the results of calculations like this by dividing out on a

calculator. In this case, the quotient is 2.892892892..., the original number in decimal form.

9. We can be certain that this decimal expansion, which we have been told is an endless
   string of digits, has a repeating pattern. The original quotient is a rational number by
   definition. Remember, any rational number can be expressed as either a terminating
   decimal or an endless repeating decimal. The repeating pattern of digits in the decimal
   expansion might be incredibly long, but it is finite.


10. This is one of the most baffling problems in mathematics. The trouble comes up
    because we’re trying to compare hard reality with pure theory. Even the most powerful
    supercomputer can be confused by a string-of-digits problem if the repeating pattern is
    complicated enough. But the fact that a pattern can’t be discovered in a human lifetime
    does not prove conclusively that there is not a pattern! It works the other way, too. If we
    see a long string of digits repeating many times, we can’t be sure it will repeat endlessly,
    unless we know that there’s a ratio of integers with the same value.